Our goal in this paper is to propose and study a LASSO variant of the
MARS method where we consider all the basis functions of the form
(REF ) and apply the LASSO method of
[1]}. As is well-known, LASSO is an
attractive alternative to stepwise regression methods in usual linear
models. In order to apply LASSO in the MARS setting, we first
assume that the explanatory variables \(x_1, \dots , x_d\)  all take
values in the interval \([0, 1]\) . In other words, the domain of the
regression function is assumed to be \([0, 1]^d\) . In a practical
setting, this can be achieved by subtracting the minimum possible
value and dividing by the range for each explanatory variable. This
scaling puts all the explanatory variables on a comparable footing
enabling the application of LASSO. Without such a scaling,
\(x_1, \dots , x_d\)  might be on very different scales in which case
penalizing or constraining the sum of the absolute values of the
coefficients corresponding to the terms \((x_j - t_j)_+\)  would be
unnatural (e.g., think of the setting where \(x_1\)  is years of
experience and \(x_2\)  is days of education). After
fitting a function in the transformed domain \([0, 1]^d\) , we can simply
invert the transformation to find the equation of the fitted function in
the original domain (see Section REF  for two real data
applications).
